How do I find the points on the ellipse #4x^2 + y^2 = 4# that are furthest from #(1, 0)#?

1 Answer

Explanation:

The ideas explained below are based in the hypothesis that the involved functions are twice continuous.

The problem can be stated as: given a set of points #(x,y)# defined through the relationship #f(x,y) = 4x^2+y^2-4=0#, and a fixed point #q=(x_q,y_q)#, determine a point #p_o=(x_o, y_o)# in the set defined by #f(x,y)=0# such that the distance #d(x,y,x) = norm(p_o-q)# has maximum value.

If such point exists, then the normal to #f(x,y)=0# in #p_o# must be linearly dependent with the normal to #d(x,y,z)# in #p_o#. In other words, the two normals must be co-lineal. Formally this condition is equivalent to